In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Final

5

By(3),(8),weeasilysee:

∞

Fq(t,x)=[2]q

(9)

k=0

1

1+qj k.

k

t

k!

Di erentiatingbothsideswithrespecttotin(5),(6)andcomparingcoe cients,

weobtainthefollowing:

Theorem1.Form≥0,wehave(10)

( m,1)Em,q(x)=[2]q

∞

n

q nm[n+x]mq( 1).

n=0

Corollary2.Letm∈N.Thenthereexists(11)

( m,1)

Em,q=[2]q

∞

n

q nm[n]mq( 1),andE0,q

(0,1)

=

[2]q

n=1

[2]qd[2]qd

[d]mq[d]mq

d 1 i=0d 1 i=0

q mi( 1)iχ(i)

q dx(m+1)[

( m,1)

i

Zp

χ(i)( 1)iq miEm,qd(

i